Ω(u,u′) =cq 1 c′p 1 −cp 1 c′q 1 +···+cqdc′pd−cpdc′qd=
(
cq 1 cp 1 ... cqd cpd)
0 1 ... 0 0
−1 0 ... 0 0
..
.
..
.
..
.
..
.
0 0 ... 0 1
0 0 ... −1 0
c′q 1
c′p 1
..
.
c′qd
c′pd
Matricesg∈M(2d,R) such that
gT
0 1 ... 0 0
−1 0 ... 0 0
..
.
..
.
..
.
..
.
0 0 ... 0 1
0 0 ... −1 0
g=
0 1 ... 0 0
−1 0 ... 0 0
..
.
..
.
..
.
..
.
0 0 ... 0 1
0 0 ... −1 0
make up the groupSp(2d,R) and preserve Ω, satisfying
Ω(gu,gu′) = Ω(u,u′)This choice of Ω is much less arbitrary than it looks. One can show that
given any non-degenerate antisymmetric bilinear form onR^2 da basis can be
found with respect to which it will be the Ω given here (for a proof, see [8]).
This is also true if one complexifiesR^2 d, using the same formula for Ω, which
is now a bilinear form onC^2 d. In the real case the group that preserves Ω is
calledSp(2d,R), in the complex caseSp(2d,C).
To get a fermionic analog of this, all one needs to do is replace “non-
degenerate antisymmetric bilinear form Ω(·,·)” with “non-degenerate symmetric
bilinear form (·,·)”. Such a symmetric bilinear form is actually something much
more familiar from geometry than the antisymmetric case analog: it is just a
notion of inner product. Two things are different in the symmetric case:
- The underlying vector space does not have to be even dimensional, one
can takeM=Rnfor anyn, includingnodd. To get a detailed analog of
the bosonic case though, we will need to consider the even casen= 2d. - For a given dimensionn, there is not just one possible choice of (·,·) up
to change of basis, but one possible choice for each pair of non-negative
integersr,ssuch thatr+s=n. Givenr,s, any choice of (·,·) can be put